Physics Letters B 688 (2010) 224–229

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Generalised uncertainty principle, remnant mass and singularity problemin black hole thermodynamics

Rabin Banerjee, Sumit Ghosh ∗

S. N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata-700098, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 February 2010Received in revised form 26 March 2010Accepted 2 April 2010Available online 9 April 2010Editor: T. Yanagida

Keywords:Generalised uncertainty principleBlack hole thermodynamics

We have derived a new generalised uncertainty principle (GUP) based on certain general assumptions.This GUP is consistent with predictions from string theory. It is then used to study Schwarzschildblack hole thermodynamics. Corrections to the mass-temperature relation, area law and heat capacityare obtained. We find that the evaporation process stops at a particular mass, referred as the remnantmass. This is instrumental in bypassing the well-known singularity problem that occurs in a semiclassicalapproach.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

The introduction of gravity into quantum field theory bringsan observer independent minimum length scale in the picture[1]. A minimal length also occurs in string theory [2], non-commutative geometry [3] or can be obtained from gedankenexperiment [4]. This minimal length is expected to be close orequal to Planck length (L P ). The manifestation of the inclusionof a minimal length in theories has been observed from differentperspectives – the generalised uncertainty principle (GUP), modi-fied dispersion relation (MDR), deformed special relativity (DSR),to name a few. The modification or deformation affect the well-known semiclassical laws of black hole thermodynamics [5–8].For instance, the black hole entropy is no longer proportional tothe horizon area [9,10]. Another interesting result is the existenceof a remnant mass of a black hole. The existence of a remnantmass of a black hole is verified by different approaches – using ageneralised uncertainty principle [11] or analysing the tunnellingprobability [12].

In this Letter we study the modifications to the laws ofSchwarzschild black hole thermodynamics by starting from a newGUP which is derived from certain basic assumptions. The con-sequences of these modifications are investigated in details. Weshow the existence of a critical (ans also singular) mass for theblack hole below (at) which the thermodynamic entities becomecomplex (ill defined). However both the critical and singular mass

* Corresponding author.E-mail addresses: rabin@bose.res.in (R. Banerjee), sumit.ghosh@bose.res.in

(S. Ghosh).

0370-2693/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2010.04.008

are less than another mass – the remnant mass. Our analysis re-veals that the black hole evaporation does not lead to a singularity.This process terminates at a finite mass which we call the rem-nant mass. Since, as already stated, both the critical and singularmasses are less than the remnant mass, the problematic situationsare avoided.

Section 2 gives the derivation of the GUP. Corrections inducedby this GUP to the black hole thermodynamics are given in Sec-tion 3. The connection of remnant mass with the singularity prob-lem is discussed in Section 4. Section 5 contains the discussions.

2. A generalised uncertainty principle (GUP)

A particle with energy close to the Planck energy E P will dis-turb the space–time significantly at least upto a length of the orderof the Planck length. It is very natural to take the metric to be afunction of the particle energy1 [14]. One can find the explicit de-pendence by solving corresponding Einstein equation where theright-hand side is given by the energy–momentum tensor of theparticle. If we assume that the particle field is a linear superposi-tion of plane wave solutions (∼ eikμxμ ), then one can easily guessthat on quantisation the particle momentum (p) and energy (E)

may be non-linear in wave vector (k) and angular frequency (ω)

[14,15]. In general we may write

kμ = f(

pμ)

(1)

It is easy to show that both kμ and pμ can transform likea Lorentz vector only for special types of function f . The stan-

1 This is the effect of back reaction [13].

R. Banerjee, S. Ghosh / Physics Letters B 688 (2010) 224–229 225

dard form is the obvious pμ = h̄kμ and a more general form ispμ = φ(kνkν)kμ where φ(kνkν) is a scalar function of the invariant(kνkν); the more general form is clearly equivalent to generalisingPlanck’s constant to a function. For simplicity in this Letter we willforego Lorentz invariance and consider the following relations [16],

k = f (p), ω = f (E) (2)

The function f satisfies certain properties [14,17]:

1. The function ( f ) and its inverse ( f −1) have to be an odd func-tion to preserve parity.

2. For small momentum/energy (E � E P ) the function should bechosen to satisfy the relationship p = h̄k.

3. We assume the existence of a minimum length, identified asthe Planck length (L P ) [1,14,17] that cannot be resolved. Sothe wave vector k = f (p) should have an upper bound 2π

L P.

Since the wave vector k = f (p) shows a saturation with re-spect to the momentum p, the momentum p = f −1(k) will bea monotonically increasing function of k.

We also assume that the commutation relations

[x,k] = i,[x, p(k)

] = i∂ p

∂k(3)

hold which lead to a uncertainty relation [18]

�p�x �∣∣∣∣⟨

1

2[x, p]

⟩∣∣∣∣ = 1

2

∣∣∣∣⟨∂ p

∂k

⟩∣∣∣∣ (4)

Observe that we are not using the field theory commutator be-tween the field and its conjugate momentum. Rather our analysisis based on the algebra (3) which is plausible.

The properties of the function f (p), enlisted below (2) cannotbe satisfied by a finite order polynomial. A possible choice is

k = f (p) = 1

L P

∞∑i=0

ai(−1)i(

L P p

h̄

)2i+1

(5)

Only odd powers of p appear in the polynomial ensuring thatf (p) is odd in p (property 1). The coefficients {ai} are all posi-tive with a0 = 1 (to satisfy p = h̄k at small energy (property 2)).The factor (−1)i ensures saturation (property 3). The third prop-erty further implies that for p → ∞, k → 2π

L P, i.e.

∞∑i=0

ai(−1)i(

L P p

h̄

)2i+1

→ 2π (6)

From (5) we get

∂k

∂ p= 1

h̄

∞∑i=0

ai(2i + 1)(−1)i(

pL P

h̄

)2i

(7)

Inverting this we obtain

∂ p

∂k= h̄

∞∑i=0

a′i

(pL P

h̄

)2i

(8)

where the new coefficients of expansions {a′i} are functions of {ai}.

It is very easy to show that the first coefficient of this invertedseries will be inverse of a0, i.e. 1.

Hence the GUP following from (4) takes the form,

�x�p �⟨

h̄

2

∞∑i=0

a′i

(L P p

h̄

)2i⟩

� h̄

2

∞∑a′

i

(L P

h̄

)2i((�p)2 + 〈p〉2)i

(9)

i=0

where we have used 〈p2i〉 � 〈p2〉i . For minimum position uncer-tainty we put 〈p〉 = 0 and our GUP becomes

�x�p � h̄

2

∞∑i=0

a′i

(L P �p

h̄

)2i

(10)

Note that all a′i ’s are positive. If only the first two terms in (10)

are considered we reproduce the GUP predicted by string theory[4,19].

3. Thermodynamics of Schwarzschild black hole with corrections

The object of this section is to use the GUP (10) to evaluatedifferent thermodynamic entities of a Schwarzschild black hole andthereby find relations among them.

Let us consider a Schwarzschild black hole with mass M. Let apair (particle–antiparticle) production occur near the event hori-zon. For simplicity we consider the particles to be massless.2 Theparticle with negative energy falls inside the horizon and that withpositive energy escapes outside the horizon and observed by someobserver at infinity. The momentum of the emitted particle (p),which also characterises its temperature (T ),3 is of the order of itsmomentum uncertainty (�p) [11]. Consequently

T = �pc

kB(11)

For thermodynamic equilibrium, the temperature of the particlegets identified with the temperature of the black hole itself.

It is now possible to relate this temperature with the mass (M)of the black hole by recasting the GUP (10) in terms of T and M.In that case the GUP has to be saturated

�p�x = ε1h̄

2

∞∑i=0

a′i

(�pL P

h̄

)2i

(12)

where the new dimensionless parameter ε1 is a scale factor satu-rating the uncertainty relation. We can later adjust it by calibratingwith some known result. We add that the product of �x and �pmay be arbitrarily large but we assume that the lower limit can beachieved.

Near the horizon of a black hole the position uncertainty ofa particle will be of the order of the Schwarzschild radius of theblack hole [6,11],

�x = ε22GM

c2(13)

The new dimensionless parameter ε2 is introduced as a scale factorand will be calibrated soon.

Substituting the values of �p (11) and �x (13) in (10), the GUPis recast as

M = εM P

4

∞∑i=0

a′i

(kB T

M P c2

)2i−1

(14)

(where we have used the relations ε = ε1ε2

, M P = L P c2

G and ch̄L P

=M P c2, M P being the Planck mass).

In the absence of correction due to quantum gravity effects onlya′

0 = 1 will survive and we should reproduce the semi classicalresult. In this approximation (14) reduces to

M = εM2

P c2

4kB T(15)

2 For massive particle the expression for temperature (11) will be modified.3 For simplicity we consider the emitted spectrum to be thermal.

226 R. Banerjee, S. Ghosh / Physics Letters B 688 (2010) 224–229

This will fix ε . Comparison with the standard semi classical

Hawking temperature [9] (T H = M2P c2

8π MkB) yields ε = 1

2π .So the mass temperature relationship is

M = M P

8π

∞∑i=0

a′i

(kB T

M P c2

)2i−1

(16)

The heat capacity of the black hole, by definition, is given by

C = c2 dM

dT(17)

Therefore from (16) we find that

C = kB

8π

∞∑i=0

a′i(2i − 1)

(kB T

M P c2

)2i−2

(18)

The nature of the heat capacity will become more illuminatingif we express it in terms of particle energy E = kB T and Planckenergy E P = M P c2. Then

C = kB

8π

∞∑i=0

a′i(2i − 1)

(E

E P

)2i−2

(19)

For E � E P the first term will predominate, and since it iswith a negative signature the heat capacity will also be negative inthis region. The heat capacity increases monotonically as E → E P .There will be a point at which the heat capacity vanishes. Weconsider the corresponding temperature to be the maximum tem-perature attainable by a black hole during evaporation. The processstops thereafter.

So a Schwarzschild black hole with a finite mass and tempera-ture, by radiation process, loses its mass and in turn its tempera-ture increases. This state corresponds to a negative heat capacity.Then it attains a temperature at which dM

dT becomes zero (zeroheat capacity), i.e. there will be no further change of black holemass with its temperature. The radiation process ends here with afinite remnant mass with a finite temperature.

One can also determine the black hole entropy (S) in a similarway. According to the first law of black hole thermodynamics it isgiven by

S =∫

c2 dM

T(20)

For technical simplification this definition is expressed in terms ofthe heat capacity (17). Then exploiting (18) and carrying out theabove integration we finally obtain

S =∫

C dT

T= kB

16π

[(M P c2

kB T

)2

+ a′1 ln

(kB T

M P c2

)2

+∞∑

i=2

a′i(2i − 1)

(i − 1)

(kB T

M P c2

)2(i−1)]

(21)

If we want to express the heat capacity and the entropy interms of the mass we have to obtain an expression for T 2 in termsof M . We can do this by squaring (16)(

8π M

M P

)2

=(

M P c2

kB T

)2

+ 2a′1 + (

a′1

2 + 2a′2

)( kB T

M P c2

)2

+ 2(a′

1a′2 + a′

3

)( kB T

M P c2

)4

+ · · · (22)

Then considering a finite number of terms, dictated by the orderof the approximation, we can obtain an expression for ( kB T

M P c2 )2 interms of M by inverting (22).

3.1. First order correction

We will next discuss the effect of first order correction. Con-sequently we can neglect the contribution of ( kB T

M P c2 )2 and higherorder terms in (22). Then we get(

kB T

M P c2

)2

= 1

( 8π MM P

)2 − 2a′1

(23)

The critical mass below which the temperature becomes a com-plex quantity is given by

Mcr =√

2a′1

8πM P (24)

We will soon show that the evaporation process terminates with amass greater than this.

Substituting (23) in (21) we get

S

kB= SBH

kB− 2a′

1

16π− a′

1

16πln

(SBH

kB− 2a′

1

16π

)− a′

1

16πln(16π)

(25)

where SBH = kB4π M2

M2P

is the Bekenstein–Hawking entropy.

We now obtain the area theorem from Eq. (25). This theoremwill appear more tractable if we introduce a new variable A′ (re-duced area) defined as

A′ = 16πG2M2

c4− 2a′

1

4π

G2M2P

c4= A − 2a′

1

4πL2

P (26)

where A is the usual area of the horizon.In terms of the reduced area the expression for entropy takes a

familiar form

S

kB= A′

4L2P

− a′1

16πln

(A′

4L2P

)− a′

1

16πln(16π) (27)

This is the area theorem in presence of the GUP (10) upto the firstorder correction. The usual Bekenstein–Hawking semiclassical arealaw is reproduced for a′

1 = 0.The important feature of (27) is that entropy is explicitly ex-

pressed as a function of the reduced area and not the actual area[5,6]. It has a singularity at zero reduced area which correspondsto a singular mass given by

Msing =√

2a′1

8πM P (28)

We will subsequently prove that the reduced area is always pos-itive in presence of quantum gravity effect and the singularity isthereby avoided during the evaporation process. Observe that, tothis order, the critical mass (24) and the singular mass (28) areidentical.

The variation of temperature and entropy with mass for differ-ent values of a′

1 is shown in Figs. 1 and 2. The interesting factabout these curves is their termination at some finite mass fornon-zero a′

1. We will explain the reason in the next section.

3.2. Second order correction

In this subsection the various thermodynamic variables arecomputed upto second order. This implies that terms upto ( kB T

M P c2 )2

in (22) are retained. With a simple rearrangement one can easilyobtain

R. Banerjee, S. Ghosh / Physics Letters B 688 (2010) 224–229 227

Fig. 1. Temperature – mass curve with a′1 = 1 (blue), a′

1 = .75 (red), a′1 = 1

3 (green)and a′

1 = 0 (black) [Semiclassical]. (For interpretation of the references to colour inthis figure legend, the reader is referred to the web version of this Letter.)

Fig. 2. Entropy – mass curve with a′1 = 1 (blue), a′

1 = .75 (red), a′1 = 1

3 (green) anda′

1 = 0 (black) [Semiclassical]. (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this Letter.)

(kB T

M P c2

)2

=[( 8π M

M P)2 − 2a′

1] ±√

[( 8π MM P

)2 − 2a′1]2 − 4(a′2

1 + 2a′2)

2(a′21 + 2a′

2)(29)

Only the (−) sign is acceptable from the (±) part, because the(+) sign will not produce the semi classical result if we put a′

1 =a′

2 = 0.At a first glance, one observes that the first order expression

for temperature (23) cannot be retrieved simply by putting a′2 = 0.

Instead, one has to put a′ 21 + 2a′

2 = 0, because both a′ 21 and 2a′

2bear the signature of the second order approximation (see (22)). Tomake this retrieval simpler we will rearrange (29) using binomialexpansion.

(kB T

M P c2

)2

= 1

( 8π MM P

)2 − 2a′1

[1 + (a′2

1 + 2a′2)

[( 8π MM P

)2 − 2a′1]2

+ · · ·]

(30)

From this expression it is clear that the first order expressionfor temperature can be obtained from the second order expressionby putting a′ 2

1 +2a′2 = 0. The critical mass is found by equating the

discriminant of (29) to zero. It is given by(8π Mcr

M P

)2

= 2a′1 + 2

√a′2

1 + 2a′2 (31)

which is greater than (24) with first order correction (since a′1, a′

2are all positive).

We conclude this section by computing the corrections (uptosecond order) to the entropy and the area law. The corrected en-tropy follows from (21)

S = kB

16π

[(M P c2

kB T

)2

+ a′1 ln

(kB T

M P c2

)2

+ 3a′2

(kB T

M P c2

)2](32)

Substituting (30) in (32) we get

S

kB=

(SBH

kB− 2a′

1

16π

)− a′

1

16πln

(SBH

kB− 2a′

1

16π

)

+∞∑j=0

c j(a′

1,a′2

)( SBH

kB− 2a′

1

16π

)− j

− a′1

16πln(16π) (33)

where SBH = kB4π M2

M2P

is the Bekenstein–Hawking entropy and

these new coefficients of expansion c j are functions of a′1 and a′

2.The coefficients c j will have an explicit dependence on a′ 2

1 + 2a′2,

which follows from (30). Hence we can get our first order result(25) by putting a′ 2

1 + 2a′2 = 0.

We can now obtain our area theorem in terms of the reducedarea (26) from (33),

S

kB= A′

4L2P

− a′1

16πln

(A′

4L2P

)+

∞∑j=0

c j(a′

1,a2)( A′

4L2P

)− j

− a′1

16πln(16π) (34)

This is the area theorem with second order correction. The expres-sion looks like the standard corrected area theorem [5,6], with therole of the actual area (A) being played by the reduced area (A′).

Some comments are in order. The singularity is again at zeroreduced area, corresponding mass being given by (28). As shownin the next section, this singular mass is also less than the remnantmass with second order correction.

4. Remnant mass and singularity problem

As discussed earlier here we show that the black hole evapora-tion terminates at a finite mass which is greater than the eitherthe critical mass Mcr (24), (31) or the singular mass (28). Thisdemonstrates the internal consistency of our calculation scheme.Consequently the usual singularity problem whereby the tempera-ture blows up, is avoided.

4.1. First order correction

Considering the first two terms in the series expansion for theheat capacity (18), we obtain

C = kB[−

(M P c2 )2

+ a′1

](35)

8π kB T

228 R. Banerjee, S. Ghosh / Physics Letters B 688 (2010) 224–229

Fig. 3. Heat capacity – mass curve with a′1 = 1 (blue), a′

1 = .75 (red), a′1 = 1

3 (green)and a′

1 = 0 (black) [Semiclassical]. (For interpretation of the references to colour inthis figure legend, the reader is referred to the web version of this Letter.)

Substituting the value of T 2 from (23)

C = kB

8π

[−

((8π M

M P

)2

− 2a′1

)+ a′

1

](36)

The variation of heat capacity with mass for different values ofa′

1 is shown in Fig. 3.The collapse of the black hole is terminated when the heat

capacity becomes zero. The mass of the black hole now remainsunchanged. This mass is called the remnant mass. Its value is ob-tained by solving

kB

8π

[−

((8π M

M P

)2

− 2a′1

)+ a′

1

]= 0 (37)

leading to,

Mrem =√

3a′1

8πM P (38)

Alternatively the value for remnant mass can also be obtainedby minimising the entropy (28),

dS

dM= 0 (39)

and looking at the second derivative ( d2 SdM2 > 0). The result (38) is

reproduced.The most important fact is that the value (38) is greater than

the singular mass (28) and also the critical mass (24). So we cansay that the singularity will be avoided during the evaporation pro-cess and the reduced area will be positive. At the same time wealso managed to avoid any possibility of dealing with complex val-ues for the thermodynamic entities (since, Mrem > Mcr ).

The remnant value of area, reduced area, temperature and en-tropy are now expressed in terms of the coefficient a′

1.

Arem = 16πG2M2rem

c4= 3a′

1

64π2

16πG2M2P

c4(40)

A′rem = a′

1L2P = 16πG2M2

rem4

= Amin (41)

4π 3c 3

Trem = 1√a′

1

M P c2

kB(42)

Srem = kB

16π

[a′

1 − a′1 ln

(a′

1

)](43)

The expressions for different thermodynamic entities includingthe remnant mass involves only one parameter a′

1 as a measureof quantum gravity effect. If we put a′

1 = 0 we get back all oursemi classical results and the remnant mass becomes zero. Thefinal entropy and heat capacity become zero while the temperaturebecomes infinite.

4.2. Second order correction

The heat capacity with the second order contribution is givenby

C = kB

8π

[−

(M P c2

kB T

)2

+ a′1 + 3a′

2

(kB T

M P c2

)2](44)

The expression for remnant mass can be found either from zeroheat capacity condition or by minimising the entropy. Adopting thefirst approach we will compute the remnant mass here. Replacingthe value of T 2 from (29) in (44) and equating the r.h.s. to zero weobtain the remnant mass(

8π Mrem

M P

)2

= 1

6a′2

[−a′1

(a′2

1 − 13a′2

)+ (

a′21 + 5a′

2

)√a′2

1 + 12a′2

](45)

One can easily show that the r.h.s. is greater than 3a′1 which is

the value (38) for ( 8π MremM P

)2 with first order correction. The rem-nant mass is also greater than the critical mass (31) for positivea′

1 and a′2. The difference between remnant mass and the critical

mass for different values of a′1 and a′

2 is shown in Fig. 4.

5. Discussion

The laws of black hole thermodynamics are known to be mod-ified by the presence of a generalised uncertainty principle (GUP)[5–8]. Here we have derived a new GUP (based on the presence ofa minimal length scale (L P )) which, at the lowest orders, is alsoshown to be compatible with string theory predictions. Using thisGUP various aspects of Schwarzschild black hole thermodynamicswere examined.

Our calculations were performed upto two orders (in L P ) of cor-rections. Corrected structures of mass-temperature relation, areatheorem and heat capacity were obtained. The usual semiclassicalexpressions were easily derived.

An important consequence was that the black hole evaporationterminated at a finite mass. This (remnant) mass was found to begreater than either the critical mass (below which the thermody-namic variables become complex) or the singular mass (where thethermodynamic variables become infinite). Consequently the ill de-fined situations were bypassed. Also, contrary to standard results[5,6] using GUP our modified area law (27), (34) is more transpar-ent when expressed in terms of reduced area defined in (26).

To put our results in a proper perspective let us compare withearlier findings. A remnant mass was also found in [11] usingstringy GUP and in [12] employing notions of tunnelling. In thefirst case the remnant mass was given by M = M P (which is con-sistent with our findings) and successfully avoided the singularity.However, in contrast to our analysis, the calculations were con-fined to the leading order only. Also, the remnant and the critical

R. Banerjee, S. Ghosh / Physics Letters B 688 (2010) 224–229 229

Fig. 4. Difference between the remnant mass and the critical mass (Mrem − Mcr) in unit of M P for different values of a′1 and a′

2 shown from two different angles. Thedifference becomes zero only at a′

2 = 0. This point is not considered, since for a′2 = 0, second order correction is not meaningful.

mass became identical. Hence it was not possible to distinguishbetween the termination of black hole evaporation and complex-ification of thermodynamic variables. The calculation of [12], onthe other hand, led to the result that although there was a rem-nant mass, the singularity problem could not be avoided.

Acknowledgements

One of the authors (S.G.) thanks the Council of Scientific andIndustrial Research (C.S.I.R), Government of India, for financial sup-port. We also thank B.R. Majhi and S.K. Modak for discussions.

References

[1] G. Amelino-Camelia, Int. J. Mod. Phys. D 11 (2002) 35, arXiv:gr-qc/0012051;G. Amelino-Camelia, Phys. Lett. B 510 (2001) 255, arXiv:hep-th/0012238v1.

[2] D. Amati, M. Ciafaloni, G. Veneziano, Phys. Lett. B 216 (1989) 41.[3] F. Girelli, E.R. Livine, D. Oriti, Nuclear Physics B 708 (2005) 411, arXiv:gr-

qc/0406100.

[4] F. Scardigli, Phys. Lett. B 452 (1999) 39.[5] G. Amelino-Camelia, M. Arzano, Y. Ling, G. Mandanici, Class. Quantum Grav. 23

(2006) 2585, arXiv:gr-qc/0506110.[6] A.J.M. Medved, E.C. Vagenas, Phys. Rev. D 70 (2004) 124021.[7] K. Nouicer, Phys. Lett. B 646 (2007) 63;

Y.M. Kim, Y.J. Park, Phys. Lett. B 655 (2007) 172.[8] A. Bina, S. Jalalzadeh, A. Moslehi, Phys. Rev. D 81 (2010) 023528, arXiv:

1001.0861.[9] S.W. Hawking, Nature (London) 248 (1974) 30;

S.W. Hawking, Commun. Math. Phys. 43 (1975) 199.[10] J.D. Bekenstein, Phys. Rev. D 7 (8) (1973) 2333.[11] R.J. Adler, P. Chen, D.I. Santiago, Gen. Rel. Grav. 33 (2001) 2101, arXiv:gr-qc/

0106080.[12] L. Xiang, Phys. Lett. B 647 (2007) 207.[13] C.O. Loustó, N. Sánchez, Phys. Lett. B 4 (1988) 411.[14] S. Hossenfelder, Phys. Rev. D 73 (2006) 105013, arXiv:hep-th/0603032.[15] João Magueijo, Phys. Rev. D 73 (2006) 124020.[16] S. Hossenfelder, Class. Quantum Grav. 23 (2006) 1815, arXiv:hep-th/0510245.[17] S. Hossenfelder, Class. Quantum Grav. 25 (2008) 038003, arXiv:0712.2811.[18] See for example David J. Griffiths, Introduction to Quantum Mechanics, Prentice

Hall, 1994, Section 3.4.[19] M. Maggiore, Phys. Lett. B 304 (1993) 65.